2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 4.3s

Alternatives: 4

Speedup: 1.2×

Specification

Math

\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.16666666666666666 \cdot \pi\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma -0.3333333333333333 (acos (/ (- g) h)) (* -0.16666666666666666 PI)))
  2.0))

C

double code(double g, double h) {
	return sin(fma(-0.3333333333333333, acos((-g / h)), (-0.16666666666666666 * ((double) M_PI)))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(-0.16666666666666666 * pi))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(-0.16666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

3. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333 - \pi \cdot 0.6666666666666666\right) + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right) + \pi \cdot \frac{1}{2}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right)} + \pi \cdot \frac{1}{2}\right) \]

   3. associate-+l- N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)} \]

   4. sub-flip N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \color{blue}{-\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\color{blue}{\pi \cdot \frac{2}{3}} - \pi \cdot \frac{1}{2}\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\pi \cdot \frac{2}{3} - \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

   10. distribute-lft-out-- N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   12. metadata-eval 100.0%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot \color{blue}{0.16666666666666666}\right)\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\pi \cdot \frac{1}{6}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\pi \cdot \frac{1}{6}\right)\right) \cdot 2} \]

   3. lower-*.f64 100.0%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot 0.16666666666666666\right)\right) \cdot 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \left(-\pi \cdot \frac{1}{6}\right)\right)} \cdot 2 \]

   5. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} + \left(-\pi \cdot \frac{1}{6}\right)\right) \cdot 2 \]

   6. lower-fma.f64 100.0%

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -\pi \cdot 0.16666666666666666\right)\right)} \cdot 2 \]

   7. lift-neg.f64 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\mathsf{neg}\left(\pi \cdot \frac{1}{6}\right)}\right)\right) \cdot 2 \]

   8. lift-*.f64 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{6}}\right)\right)\right) \cdot 2 \]

   9. *-commutative N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{neg}\left(\color{blue}{\frac{1}{6} \cdot \pi}\right)\right)\right) \cdot 2 \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot \pi}\right)\right) \cdot 2 \]

   11. lower-*.f64 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot \pi}\right)\right) \cdot 2 \]

   12. metadata-eval 100.0%

       \[\leadsto \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{-0.16666666666666666} \cdot \pi\right)\right) \cdot 2 \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.16666666666666666 \cdot \pi\right)\right) \cdot 2} \]
8. Add Preprocessing

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[2 \cdot \sin \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 0.5235987755982989\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (- (* -0.3333333333333333 (acos (/ (- g) h))) 0.5235987755982989))))

C

double code(double g, double h) {
	return 2.0 * sin(((-0.3333333333333333 * acos((-g / h))) - 0.5235987755982989));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = 2.0d0 * sin((((-0.3333333333333333d0) * acos((-g / h))) - 0.5235987755982989d0))
end function

Java

public static double code(double g, double h) {
	return 2.0 * Math.sin(((-0.3333333333333333 * Math.acos((-g / h))) - 0.5235987755982989));
}

Python

def code(g, h):
	return 2.0 * math.sin(((-0.3333333333333333 * math.acos((-g / h))) - 0.5235987755982989))

Julia

function code(g, h)
	return Float64(2.0 * sin(Float64(Float64(-0.3333333333333333 * acos(Float64(Float64(-g) / h))) - 0.5235987755982989)))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * sin(((-0.3333333333333333 * acos((-g / h))) - 0.5235987755982989));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5235987755982989), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

3. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333 - \pi \cdot 0.6666666666666666\right) + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right) + \pi \cdot \frac{1}{2}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right)} + \pi \cdot \frac{1}{2}\right) \]

   3. associate-+l- N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)} \]

   4. sub-flip N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \color{blue}{-\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\color{blue}{\pi \cdot \frac{2}{3}} - \pi \cdot \frac{1}{2}\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\pi \cdot \frac{2}{3} - \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

   10. distribute-lft-out-- N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   12. metadata-eval 100.0%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot \color{blue}{0.16666666666666666}\right)\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot 0.16666666666666666\right)\right)} \]

6. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \color{blue}{-0.5235987755982989}\right)\right) \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \frac{-2358079250676147}{4503599627370496}\right)} \]

   2. add-flip N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \left(\mathsf{neg}\left(\frac{-2358079250676147}{4503599627370496}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} - \left(\mathsf{neg}\left(\frac{-2358079250676147}{4503599627370496}\right)\right)\right) \]

   4. lower--.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{-2358079250676147}{4503599627370496}\right)\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} - \left(\mathsf{neg}\left(\frac{-2358079250676147}{4503599627370496}\right)\right)\right) \]

   6. metadata-eval 98.5%

      \[\leadsto 2 \cdot \sin \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{0.5235987755982989}\right) \]

8. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right) - 0.5235987755982989\right)} \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 1.2× speedup

Math

\[2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.5235987755982989\right)\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (fma (acos (/ (- g) h)) -0.3333333333333333 -0.5235987755982989))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(acos((-g / h)), -0.3333333333333333, -0.5235987755982989));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(acos(Float64(Float64(-g) / h)), -0.3333333333333333, -0.5235987755982989)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333 + -0.5235987755982989), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

3. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333 - \pi \cdot 0.6666666666666666\right) + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right) + \pi \cdot \frac{1}{2}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \pi \cdot \frac{2}{3}\right)} + \pi \cdot \frac{1}{2}\right) \]

   3. associate-+l- N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} - \left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)} \]

   4. sub-flip N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \mathsf{neg}\left(\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)\right)\right)\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \color{blue}{-\left(\pi \cdot \frac{2}{3} - \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\color{blue}{\pi \cdot \frac{2}{3}} - \pi \cdot \frac{1}{2}\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\left(\pi \cdot \frac{2}{3} - \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right) \]

   10. distribute-lft-out-- N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, -\color{blue}{\pi \cdot \left(\frac{2}{3} - \frac{1}{2}\right)}\right)\right) \]

   12. metadata-eval 100.0%

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot \color{blue}{0.16666666666666666}\right)\right) \]

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -\pi \cdot 0.16666666666666666\right)\right)} \]

6. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, \color{blue}{-0.5235987755982989}\right)\right) \]
7. Add Preprocessing

Alternative 4: 97.6% accurate, 1.2× speedup

Math

\[2 \cdot \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (sin (fma 0.3333333333333333 (acos (/ (- g) h)) 3.6651914291880923))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(0.3333333333333333, acos((-g / h)), 3.6651914291880923));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), 3.6651914291880923)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + 3.6651914291880923), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\sin \left(\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   5. +-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3} + \frac{2 \cdot \pi}{3}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. associate-+l+ N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3} + \left(\frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}} + \left(\frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   8. mult-flip N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{1}{3}} + \left(\frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} + \left(\frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{1}{3}}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\frac{2 \cdot \pi}{3}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   13. lift-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{\color{blue}{2 \cdot \pi}}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{\color{blue}{\pi \cdot 2}}{3} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   15. associate-/l* N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\pi \cdot \frac{2}{3}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

3. Applied rewrites 97.6%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{fma}\left(\pi, 0.6666666666666666, \pi \cdot 0.5\right)\right)\right)} \]

4. Evaluated real constant 97.6%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{3.6651914291880923}\right)\right) \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025209 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))